Data depth based support vector machines for predicting corporate bankruptcy

Abstract

In financial distress analysis, the diagnosis of firms at risk for bankruptcy is crucial in preparing to hedge against any financial damage the at-risk firms stand to inflict. Some pre-alarm signals that indicate a potential financial crisis exist when a firm faces a default risk. Early studies on corporate bankruptcy prediction include parametric and nonparametric approaches, such as artificial intelligence (AI), for detecting pre-alarm signals. Among nonparametric techniques, the methods involving support vector machine (SVM) have shown potential in predicting corporate bankruptcy. We propose a hybrid method that combines data depths and nonlinear SVM for the prediction of corporate bankruptcy. We employed data depth functions to condense multivariate financial data with nonlinear and non-normal characteristics into one-dimensional space. The SVM method was introduced to classify the data points on a depth versus depth plot (DD-plot). Based on data set that records failed and non-failed manufacturing firms in Korea over 10 years, the empirical results demonstrated that the proposed method offers a higher level of accuracy in corporate bankruptcy prediction than existing methods. The proposed method is expected to provide a guidance in corporate investing for investors or other interested parties.

1 Introduction

External uncertainties that pose a threat to corporate survival have increased drastically due to the protean environment of the global economy. In the face of such uncertainties, financial institutions are endeavoring to prepare sophisticated countermeasures against delays or defaults on liability fulfillment on the part of firms. The prediction of default risks in firms by companies that are liable to pay is the aim of financial institutions in order to minimize their capital exposure to risks and to mitigate their own default risks. The prediction of bankruptcy has been extensively studied in finance and management literature for the last two decades (Lee et al. [17]; Salcedo-Sanz et al. [32]; Min and Lee [26]; Li and Sun [18]; Tsai [35], to name a few). Bankruptcy prediction has become even more important since the Basel Committee on Banking Supervision (Basel II) established borrowers’ risk rating as a key criterion for minimum capital requirements of banks. In general, when a firm faces an inevitable default on liability fulfillment, there may exist some symptoms or pre-alarm signals indicating a financial crisis for the firm itself.

Early studies of bankruptcy prediction exploited parametric statistical techniques such as multiple discriminant analysis (MDA) [2], logit model [29] and probit model [41]. Strict assumptions for the statistical approaches (e.g., linearity, normality, and pre-existing functional forms relating criterion variables to predictor variables), however, have limited the applications of these techniques in the area of finance. To overcome this obstacle, nonparametric techniques such as artificial intelligence (AI) have been employed since the late 1980s for the prediction of corporate bankruptcy or financial distress. The AI techniques include decision trees (DTs), artificial neural networks (ANNs), genetic algorithms (GAs), and back-propagation networks (BPNs). Odom and Sharda [28] first introduced an ANN model to predict corporate bankruptcy. Tam and Kiang [34] compared the performance of a neural network model with the performances of linear discriminant models, logit models, the iterative dichotomiser 3 (ID3) algorithm, and the k-nearest neighbor approach based on insolvent data of commercial banks. They showed that the ANN provides the most accurate and robust prediction results among the various methods. While a number of application studies have reported the outstanding performance of ANNs, the models have difficulty in clearly explaining prediction results due to the lack of explanatory power and capacity for generalization because of overfitting problems. Additionally, ANNs require significant time and efforts to construct a best architecture through multiple layers [16, 33].

As an alternative to ANNs, a method based on support vector machine (SVM) has recently attracted special attention in the area of financial distress modeling. The SVM has reported better classification results than parametric statistical methods or other nonparametric techniques such as ANNs and BPNs. Moreover, SVM can overcome the overfitting problem via the concept of structural risk minimization. Häardle et al. [13] first introduced the SVM to corporate bankruptcy prediction, comparing its performance with ANN and MDA methods, as well as a learning vector quantization proposed by Fan and Palaniswami [10]. By mapping input variables onto a high-dimensional feature space, Min and Lee [26] showed that SVM transforms complex problems of corporate bankruptcy prediction into simpler ones, to which linear discriminant functions can subsequently be applied. Häardle et al. [14] explored the suitability of smooth SVMs in predicting corporate default risk. They investigated how key factors such as selection of appropriate accounting ratios, length of training period, and structure of training samples influence prediction precision for corporate bankruptcy. While SVM achieves an excellent classification accuracy, a main disadvantage of the method is difficulty in interpreting the modeling results.

In the last decade, a number of researchers have actively developed hybrid approaches to predict corporate bankruptcy. Hybrid approaches combine several classification methods to secure greater accuracy than individual (parametric or nonparametric) models. Min et al. [27] and Ahn et al. [1] employed a genetic algorithm to design an SVM-based technique for corporate bankruptcy prediction. The selection of both SVM hyper-parameters and input features was integrated into one learning process in their genetic algorithm. Van Gestel et al. [38] applied the Bayesian evidence framework [24, 37] to find hyper-parameters for the least squares SVM.

In this article, we propose a hybrid method based on data depth (DD) and SVM to improve the accuracy of bankruptcy prediction for Korean firms. As a nonparametric multivariate technique, DD estimates a representative value from multivariate data which may possess nonlinear and non-normal characteristics. The hybrid method (referred to hereafter as DD-SVM) calculates the DD for annual financial ratios because the ratios are generally unlikely to follow a multivariate normal distribution. Additionally, the method applies nonlinear SVM to the DD plot, which presents the depth values of the combined sample of both failed and non-failed firms to classify a binary output variable. The performance of DD-SVM is compared with other parametric methods or AI methods, including ANNs, in terms of bankruptcy prediction accuracy.

The remainder of this paper is organized as follows. Basic modeling ideas for predicting corporate bankruptcy are presented in Section 2. The ideas are based mainly on the introduction of nonlinear SVM to DD plots to classify failed or non-failed firms. The research data and pre-analytical results are given in Section 3. Section 4 presents an empirical analysis of corporate bankruptcy prediction to demonstrate the performance of the proposed method, along with a comparison of the proposed method with other competing prediction models. Section 5 concludes this study and discusses directions for future research.

2 The modeling of bankruptcy prediction

2.1 Data depth functions

The word “depth” was first used by Tukey [36] to depict high-dimensional data, and the far reaching ramifications of depth in ordering and analyzing multivariate data have been elaborated in the works of Liu [21], Donoho and Gasko [8], Liu et al. [22] and others. Data depth characterizes the centrality of high-dimensional data with respect to a distribution or a multivariate sample. Viewed as a method of dimension reduction, DD does not rely on link functions, kernel functions, or other refined mappings, unlike related methods such as principal components.

In order to form a general definition of a depth function, Zuo and Serfling [42] defined a statistical depth function as a bounded, non-negative mapping that satisfies four desirable properties: (1) affine invariance; (2) maximality at center; (3) monotonicity relative to the deepest point; and (4) vanishing at infinity. Affine invariance means that the relative depth of a point should not depend on the underlying coordinate system or the scales of the underlying measurements. For a distribution with a uniquely defined center, maximality at center indicates that the depth function should attain the maximum at this center. Monotonicity relative to the deepest point means that, as a point moves from the center outward, the corresponding depth should decrease monotonically. Vanishing at infinity means that the depth of a point approaches zero when its norm approaches infinity. Among a number of DD functions possessing the above properties, Mahalanobis depth, simplicial depth and Tukey’s depth functions are the most popular.

2.1.1 Mahalanobis depth

Mahalanobis [25] introduced a distance function based on Hotelling’s T ² statistic (now called “Mahalanobis depth”). Serving as the first DD concept, the Mahalonobis depth measures how deep a point $\mathbf {x} \in \mathbb {R}^{p}$ is with respect to a given distribution G. The Mahalanobis depth function is given by

$$ \mathcal{M}\mathcal{D}(G;\mathbf{x})=\frac{1}{1+(\mathbf{x}-\boldsymbol{\mu}_{G})^{T} \boldsymbol{\Sigma}_{G}^{-1}(\mathbf{x}-\boldsymbol{\mu}_{G})}, $$

(1)

where μ _G and Σ _G denote the mean vector and the covariance of the reference distribution G, respectively. In general, because G is unknown, the sample version of the Mahalanobis depth is obtained by replacing μ _G and Σ _G with their sample estimates $\bar {\mathbf {x}}_{n}$ and S _n , respectively, for multivariate data set { x ₁,…,x _n }.

2.1.2 Simplicial depth

Liu [21] introduced the concept of “simplicial depth”, which is determined by counting simplices derived from n data points. For a reference distribution G on $\mathbb {R}^{p}$, the simplicial depth of a data point x with respect to G is defined by

$$ \mathcal{SD}(G;\mathbf{x})=P_{G}\{\mathbf{x} \in S[\mathbf{x}_{1},\ldots\mathbf{x}_{p+1}]\}, $$

(2)

where x ₁,…,x _p+1 are independent observations from G and S[x ₁,…,x _p+1] is the simplex with vertices x ₁,…,x _p+1. In other words, S[x ₁,…,x _p+1] is the set of all points in $\mathbb {R}^{p}$ that are convex combinations of {x ₁,…,x _p+1}. The sample version of $\mathcal {SD}$ is obtained by replacing G in $\mathcal {SD}(G;\mathbf {x})$ with G _n , or alternatively, by computing the fraction of the sample random simplices containing the point x as

$$ \mathcal{SD}(G_{n};\mathbf{x})\!=\left( \begin{array}{c} n\\ p\,+\,1 \end{array}\right)^{-1} \sum\limits_{1 \leq i_{1} <{\cdots} <i_{p+1} \leq n}I_{(\mathbf{x}\in S[\mathbf{x}_{i_{1}},\ldots,\mathbf{x}_{i_{p+1}}])}, $$

(3)

where I _(⋅) is the indicator function. Liu [21] showed that $\mathcal {SD}(G; \mathbf {x})$ is affine invariant, and that if G is absolutely continuous, then $\mathcal {SD}(G_{n};\mathbf {x})$ converges uniformly and strongly to $\mathcal {SD}(G;\mathbf {x})$ as n → ∞. we can confirm that x is contained in the simplex $S[\mathbf {x}_{i_{1}},\ldots ,\mathbf {x}_{i_{p+1}}]$ if x can be expressed as a convex combination of {$\mathbf {x}_{i_{1}},\ldots ,\mathbf {x}_{i_{p+1}}$}.

2.1.3 Tukey depth

Tukey [36] proposed a half-space depth which is now commonly called “Tukey depth”. The half-space depth is the smallest proportion of data points contained on one side of any hyperplane passing through a data point x, including points lying on the hyperplane. That is, the Tukey depth is defined as

$$\begin{array}{@{}rcl@{}} &&\mathcal{TD}(G;\mathbf{x}) = \underset {H}{\inf } ~\{ P_{G} (H)\\&&{\kern53pt}: H \text{ is a closed half-space in } \mathbb{R}^{p} \text{ for } \mathbf{x} \in H \} \end{array} $$

with respect to the reference distribution G.

In bivariate data, for instance, the Tukey depth calculates the smallest proportion of data points contained on one side of any line (L) passing through x, including points lying on the line itself. Following a method proposed by Rousseeuw and Ruts [30], for example, two-dimensional Tukey depth requires a vector connecting a fixed x to each member of the reference sample x ₁,…,x _n and then measures the angles of these vectors with the positive x-axis. Instead of counting the minimum number of points lying on one side of the line passing through x and a reference sample, we can count the minimum number of angles between the angle of L and its opposite angle. With that, the empirical formula for the Tukey depth of x is

$$\mathcal{TD}(G_{n};\mathbf{x})=\frac{1}{n} ~ \underset{i}{\min} ~\{\min(k_{i}, n-k_{i})\},$$

where k _i = ψ ₁(i) − ψ ₂(i) for ψ ₁(i) = #{j : 0 ≤ 𝜃 _j < 𝜃 _i + π}, and ψ ₂(i) = #{j : 0 ≤ 𝜃 _j < 𝜃 _i }. Here, 𝜃 _i is the angle of u _i = (x _i −x)/∥x _i −x∥ for i = 1,…,n. We can assume 0 = 𝜃 ₁ ≤… ≤ 𝜃 _n < 2π and 𝜃 _n+1 = 𝜃 ₁ + 2π, 𝜃 _n+2,= 𝜃 ₂ + 2π, and so on. See Bae et al. [3] for the details on calculation of the Tukey depth, along with simple examples.

In addition to these three popular data depths, there are several other DD metrics, e.g., “convex hull peeling depth” by Barnett [4], “likelihood depth” by Fraiman and Meloche [11], “regression depth” by Rousseeuw and Jubert [31], and “Lens depth” by Liu and Moddares [23], to name a few.

2.2 Depth vs. depth plot

The depth vs. depth plot (DD-plot), which was first introduced by Liu et al. [22], is a useful analytical tool for graphical comparisons of two multivariate distributions or samples based on DD. For any two given multivariate samples, the DD-plot represents the depth values of the combined sample under the two corresponding empirical distributions. The tool transforms the two multivariate samples in any dimension to a simple two-dimensional scatter plot. Li et al. [19] addressed some advantages of the DD-plot in classification problems, including that the best separating curve in the DD-plot is determined automatically by the underlying probabilistic geometry of the data, and that classification outcomes can be easily visualized in two-dimensional DD-plots. This is much simpler than tracking classification outcomes in the original sample space of high-dimensional multivariate data. In particular, the DD-plot is robust against outliers and extreme values.

Let F and G be two distributions on $\mathbb {R}^{p}$, and let $\mathcal {D}$ be an affine-invariant depth. For two random samples drawn from F and G, {x ₁,…,x _n }(≡X) and {y ₁,…,y _m }(≡Y), respectively, the DD-plot is defined as

$$ \mathcal{DD}(F,G)=\{(\mathcal{D}(F;\mathbf{x}),\mathcal{D}(G;\mathbf{x})) \text{ for all } \mathbf{x} \in \mathbb{R}^{p}\}, $$

(4)

when F and G are known. If both F and G are unknown, then the sample version of DD-plot is given by

$$ \mathcal{DD}(F_{n},G_{m})=\left\{(\mathcal{D}(F_{n};\mathbf{x}),\mathcal{D}(G_{m};\mathbf{x})), \mathbf{x} \in \{\mathbf{X} \cup \mathbf{Y}\} \right\}. $$

(5)

Note that both $\mathcal {DD}(F,G)$ and $\mathcal {DD}(F_{n},G_{m})$ are always subsets of $\mathbb {R}^{2}$, no matter the size of the dimension p of the data. If the two given distributions are identical (that is, F ≡ G), then the resulting $\mathcal {DD}(F, G)$ is simply a line segment on the 45 ^∘ line in the DD-plot. Deviation patterns from this straight line indicate a specific type of difference between the two underlying distributions.

Figure 1 illustrates the DD-plot for simulated multivariate data. Figure 1a shows the DD-plot for two samples (n = m = 500) drawn from the standard bivariate normal distribution. It can be observed that the data are scattered around the 45 ^∘ line in the plot. Figure 1b presents the DD-plot for two samples (n = m = 500) with one drawn from the standard bivariate normal distribution and the other drawn from the bivariate normal with mean (2,0)^T. All of the DD-plots are constructed using the Mahalanobis depth. The DD-plot shows quite clearly that the observations from two different distributions scatter around the 45 ^∘ line in a manner that is almost symmetric. The 45 ^∘ line can thus be used as the separating line for two different samples in the DD-plot. Its corresponding classification rule is simple in that we assign x to F if $\mathcal {D}(F_{n};\mathbf {x})>\mathcal {D}(G_{m};\mathbf {x})$, and we assign x to G otherwise. Note that the classifier in the DD-plot is conceptually the same as the maximum depth classifier in the work of Ghosh and Chaudhuri [12].

Fig. 1

DD-plots for two random samples drawn from (a) an identical distribution and (b) two different distributions (The DD-plot is re-produced from the example by Li et al. [19])

In general, however, high-dimensional observations do not scatter symmetrically along a 45 ^∘ line because they have different dispersion structures as well as different locations. Hence, the linear classifier does not perform well. In this study, we introduce a nonlinear support vector machine (SVM) to classify failed firms and non-failed firms based on the DD-plot.

2.3 The nonlinear support vector machine

The prediction of bankruptcy can be formulated as a two-class classification problem. We apply the SVM approach to bankruptcy prediction using real-life data for Korean manufacturing companies, and compare the empirical results of the SVM-based method with the results from other prediction models.

A support vector machine, which was introduced from statistical learning theory by Vapnik [39], is a powerful classification method that provides better solutions to decision boundaries than can be obtained via traditional neural networks. SVM uses a linear model to construct nonlinear class boundaries through nonlinear mapping of input vectors into a high-dimensional feature space. In general, the linear model in the new space represents a nonlinear decision boundary in the original space. The SVM approach implements the principle of structural risk minimization, which aims to reduce the bounds of misclassification errors by constructing an optimal separating hyperplane in a high-dimensional feature space using quadratic programming to find unique solutions. The application areas of SVMs include text categorization, digital image identification, handwriting recognition, function approximation and regression, and time series forecasting.

Assuming that there are two-dimensional DD-predictors $(\mathcal {D}(F_{n};\mathbf {x}),\mathcal {D}(G_{m};\mathbf {x})) \equiv \mathbf {y}$ based on p financial ratios in our bankruptcy classifier, the (bivariate) data-depth data of predictor variables for the i th firm in the DD-plot can be represented by the vector y _i . The financial status of the i th firm is denoted by z _i ∈{+1,−1}, where + 1 represents a non-failed firm and − 1 represents a failed firm. Given a training set $D=\{\mathbf {y}_{i}, z_{i}\}_{i=1}^{N}$ for constructing two parallel bounding hyperplanes at opposite sides of a separating hyperplane w ^T y + b = 0, where w is the weight vector and b is the bias, the decision rule of SVM, which finds the optimal hyperplane separating the binary decision classes, is given for the linearly separable case as

$$ z(\mathbf{y}) =\text{sign} \left( \sum\limits_{i=1}^{N} z_{i}({\mathbf{y}_{i}^{T}} \mathbf{y})+b \right). $$

(6)

For the nonlinearly separable case, (6) is modified as follows:

$$ z(\mathbf{y}) =\text{sign} \left( \sum\limits_{i=1}^{N} z_{i} \mathcal{K}(\mathbf{y}_{i}, \mathbf{y})+b \right), $$

(7)

where $\mathcal {K}(\mathbf {y}_{i}, \mathbf {y})$ is defined as the kernel function which performs nonlinear mapping between input space and a (high-dimensional) feature space. Popular kernel functions to construct the decision rules include the radial basis function (RBF) kernel $\mathcal {K}(\mathbf {y}_{i},\mathbf {y})=\exp (-\sigma {\|\mathbf {y}-\mathbf {y}_{i}\|^{2}})$, where σ is a tuning parameter; the linear kernel $\mathcal {K}(\mathbf {y}_{i},\mathbf {y})={\mathbf {y}_{i}^{T}}\mathbf {y}$; the polynomial kernel $\mathcal {K}(\mathbf {y}_{i},\mathbf {y})=(\gamma +{\mathbf {y}_{i}^{T}}\mathbf {y})^{d}$ with degree d and a tuning parameter γ(≥ 0); and the multilayer perceptron (MLP) kernel $\mathcal {K}(\mathbf {y}_{i},\mathbf {y})= \tanh (\gamma _{1} + \gamma _{2} {\mathbf {y}_{i}^{T}}\mathbf {y})$. Note that the MLP kernel is not positive semi-definite for all choices of the tuning parameter γ ₁ and γ ₂.

Most of classification problems are, however, linearly (or nonlinearly) non-separable, thus it is generally not possible to find a hyperplane that can differentiate between “failed” and “non-failed” examples without mistakes. By allowing a certain level of misclassification, the soft margin method introduces slack variables measuring the degree of misclassification of y _i , and a cost constant C which weights the importance of classification errors vis-à-vis the margin width. The solution of the primal problem to find the optimal separating hyperplane is generally obtained after constructing the Lagrangian. The dual problem of finding an optimal separating hyperplane, along with the kernel function $\mathcal {K}(\mathbf {y}_{i},\mathbf {y})$, is re-written as the following quadratic programming problem:

$$\begin{array}{@{}rcl@{}} & \underset {\boldsymbol{\alpha}} {\text{arg min}} & Q(\boldsymbol{\alpha})=\frac{1}{2}\sum\limits_{i,j=1}^{N} \alpha_{i}\alpha_{j} z_{i} z_{j} \mathcal{K}(\mathbf{y}_{i},\mathbf{y}_{j})-\sum\limits_{i=1}^{N} \alpha_{i}, \\ & \text{s.t.} & \sum\limits_{i=1}^{N} \alpha_{i} z_{i}=0, ~~ 0\leq\alpha_{i}\leq C, ~~ i=1,2,\ldots,N, \end{array} $$

for the constant C with respect to the Lagrange multipliers α ≡ (α ₁,…,α _N )^T. Under the dual formulation, the two-class classification problem via implementation of the optimal separating hyperplane in the feature space is determined by the nonlinear SVM classifer as

$$ z(\mathbf{y}) =\text{sign} \left( \sum\limits_{i=1}^{N} \alpha_{i} z_{i} \mathcal{K}(\mathbf{y}_{i}, \mathbf{y})+b \right). $$

(8)

Note that data instances corresponding to non-zero α _i ’s are called support vectors. If the bias term b is implicitly a part of the kernel function, as in the case of the RBF kernel, then (8) is reduced to

$$z(\mathbf{y}) =\text{sign} \left( \sum\limits_{j=1}^{\text{number of SVs}} \alpha_{j} z_{j} \mathcal{K}(\mathbf{y}, \mathbf{y}_{j}) \right).$$

Among the big issues in SVM, the selection of appropriate values of parameters plays an important role in building a bankruptcy prediction model with high prediction accuracy and stability. The choice of the parameter C is very important. A large value of C permits the optimization to find a large margin, whereas a small value of C, which allows a small margin, results in a large number of support vectors. If the value of C is chosen to be unnecessarily large, the margin becomes narrow and the constructed classification model may fail to classify new objects properly, although the training set is separated well. Banz et al. [6] used a value of C = 5, however, there are no general rules to guarantee the best parameter values for application problems. Lin [20] provided a systematic method for selecting parameter values for SVMs by adapting the sampling theory concept into a Gaussian filter. In corporate bankruptcy prediction, Min and Lee [26] proposed a grid-search technique using five-fold cross-validation to determine optimal parameter values of C and σ in the RBF kernel function of the SVM. Min et al. [27] and Wu et al. [40] used genetic algorithm to optimize the values of the parameters C and σ with the RBF kernel in SVM. Van Gestel et al. [38] introduced Bayes’ formula to the inference of the RBF kernel parameter σ in the least squares SVM.

3 Research data

The sample of firms for analysis herein consists of manufacturing firms in the Korea Composite Stock Price Index 2000 (KOSPI 2000) from 2000 to 2013. The manufacturing firms of interest were extracted from the Korea Listed Companies Association (KLCA). The KLCA also provides financial information related to the firms through audit reports from external auditors. This research studies 144 firms that have filed bankruptcy petitions in Korea in the 21st century with the main objective of taking into account corporate management risks, and excludes cases of irresistible liability failures from structural risks such as abrupt economic crises (e.g., the bailout from the International Monetary Fund (IMF) in 1997). Originally, corporate failure has been defined as bankruptcy, default on bonds, overdrawing of a bank account, or non-payment of a preferred stock dividend [5]. For simplicity, however, we restricted the definition of corporate failure in this study to only corporate bankruptcy. For the same period, we selected 144 ‘non-failed’ firms randomly from all solvent firms. A failed firm was paired with a non-failed firm in a similar industry, which deals with similar products, with similar capitalization, and with similar values of assets. The one-to-one matching ratio is a potential cause of oversampling problems [41]. However, in order to highlight the effects of key financial ratios on the likelihood of corporate bankruptcy, a matched sample of non-failed firms was selected. We only selected medium-sized and large-sized firms with property amounts of at least $10 billion. The failed and non-failed data were arbitrarily split into two subsets: about 80% of the data was used for a training set and 20% for a validation set for k-fold cross validation. The training data was used to build the bankruptcy prediction model using the data depth and SVM. The prediction model was verified by the validation data which was not utilized to construct the model.

The KLCA reported 111 financial ratios representing profitability, stability, activity, and productivity with respect to individual firms. Out of them, we selected 53 significant ratios using two-sample t-test between failed firms and non-failed firms. The selected financial ratios, which are summarized in Table 1, were used to build a prediction model for classifying failed or non-failed firms. We analyzed the financial data on individual firms for 10 years preceeding bankruptcy (or survival up to 2013) to examine the existence of chronological trends in corporate bankruptcy.

Table 1 Financial ratios included in the bankruptcy prediction model

3.1 Multivariate normality test

Existing parametric distress models have been constructed under the assumption of (multivariate) normality. However, empirical results have shown that, in practice, most financial ratios violate the normality assumption, thereby justifying the introduction of nonparametric techniques. First, we assessed the normality of financial ratios using the Henze-Zirkler’s test statistic [15]. The Henze-Zirkler test statistic for multivariate normality is given by

$$\begin{array}{@{}rcl@{}}T_{HZ} &=& \frac{1}{n} \sum\limits_{i,j=1}^{n} e^{-\frac{\xi^{2}}{2} A_{ij}} - 2(1+ \xi^{2})^{-p/2} \sum\limits_{j=1}^{n} e^{-\frac{\xi^{2}}{2(1+ \xi^{2})} A_{j}}\\ &&+ n(1+ 2\xi^{2})^{-p/2},\end{array} $$

where p is the number of variables, $\xi = \sqrt {2}^{-1}$ (n(2p + 1)/4)^−(p+4), $A_{i} = (\mathbf {x}_{i} - \bar {\mathbf {x}})^{T} \boldsymbol {S}^{-1}(\mathbf {x}_{i} - \bar {\mathbf {x}})$ is the Mahalanobis distance of the i th observation to the centroid, A _{i j} = (x _i −x _j )^T S ⁻¹(x _i −x _j ) is the Mahalanobis distance between the i th and j th observations, and ξ is a smoothing parameter. If the data are (multivariate) normally distributed, then the test statistic is approximately log-normally distributed, with mean and variance respectively corresponding to

$$\begin{array}{@{}rcl@{}} \text{E}[T_{HZ}] &=& 1-(1+ 2\xi^{2})^{-p/2}\\ &&\times\left\{1+ \frac{p \xi^{2}}{(1+ 2\xi^{2})}+ \frac{p(p+2)\xi^{4}}{2(1+ 2\xi^{2})^{2}}\right\}, \quad \text{and} \\ \text{Var}[T_{HZ}] &=& 2(1+ 4\xi^{2})^{-p/2} +2(1+ 2\xi^{2})^{-p}\\&&\times \left\{ 1+ \frac{2 p\xi^{4}}{2(1+ 2\xi^{2})^{2}} + \frac{3p(p+2)\xi^{8}}{4(1+ 2\xi^{2})^{4}}\right\} \\ & & -4 \omega^{-p/2} \left\{1+ \frac{3p \xi^{4}}{2\omega}+ \frac{p(p+2)\xi^{8}}{2 \omega^{2}} \right\}, \end{array} $$

where ω = (1 + ξ ²)(1 + 3ξ ²). Using the log-normal distribution parameters, the Wald test statistic can be applied to test the significance of multivariate normality. The Henze-Zirkler test was implemented by an MVN package in R software for assessing multivariate normality. The results of multivariate normality testing of corporate financial ratios for 10 years are summarized in Table 2. The periods represent the years prior to bankruptcy for failed firms and survival years for non-failed firms. All of the p-values derived from the Henze-Zirkler test were very close to zero, thus leading us to conclude that the data set did not satisfy the multivariate normality assumption.

Table 2 Multivariate normality tests for financial ratios of the sampling firms in Korea

4 Analytical results

To predict the bankruptcy of manufacturing firms in Korea, this study was conducted according to the following steps:

• 1) Reduction of the number of multi-dimensional financial ratios by DD.

• 2) Plotting of the values of DD into a DD-plot.

• 3) Classification of the DD in the DD-plot using nonlinear SVM.

We compared the performance of the proposed method with other bankruptcy prediction models in the existing literature. The hit ratio of classification was used as an indicator to evaluate the predictive accuracy of each models. In addition, Type I error (defined as the probability that a firm predicted not to fail will, in fact, fail) and Type II error (defined as the probability that a firm predicted to fail will not, in fact, fail) were also included in the evaluation criteria.

4.1 Computing data depths

The preliminary analysis shows that the financial ratios of Korean manufacturing firms deviate from the multivariate normality assumption. Instead of parametric approaches, therefore, we introduce nonparametric methods based mainly on DD functions to predict bankruptcy among sample manufacturing firms in Korea. Three types of DD were applied to reduce the high-dimensional financial ratios: the Mahalanobis depth, the simplicial depth, and the Tukey depth. For failed firms x _i (i = 1,2,…,144) and non-failed firms y _i (i = 1,2,…,144), 53 financial ratios were condensed into one-dimensional measure of DD, without any distributional assumption. For example, Fig. 2 presents the depth values resulting from the Mahalanobis measure for both failed and non-failed firms for 10 years. Figure 2 does not show any remarkable difference between the two groups in a scatter plot. We could find similar trend with respect to the other two depth measures. In general, because computing the Tukey depth for dimensions higher than three is intractable using the method by Rousseeuw and Ruts [30], we can use the random approximation algorithm introduced by Cuesta-Albertos and Nieto-Reyes [7], which is computationally efficient in high-dimensional data-sets.

Fig. 2

Plot of the Mahalanobis depth values (+ : failed firms, ∘: non-failed firms)

4.2 Plotting via DD-plot

A DD-plot can serve as a simple diagnostic tool for visual comparison between two samples of any dimension. Distributional differences (e.g., changes of location, scale, skewness or kurtosis) may present different graphical patterns in DD-plots. We drew the DD-plots based on the three measures of DD. For example, the DD-plots based on the Mahalanobis depth for 10 years are given in Fig. 3. In comparing failed and non-failed firms, the scatter plots in Fig. 3a and b present any separable differences between the two groups. Note that the failed firms scatter at the bottom of the DD-plot at two years prior to bankruptcy, representing a location shift from the group of non-failed firms. However, this separation trend decreased gradually at five (Fig. 3c) and six years (Fig. 3d) prior to bankruptcy or survival. At nine (Fig. 3e) and ten years (Fig. 3f) (prior to bankruptcy or survival), the resulting $\mathcal {DD}(F, G)$ were scattered on the 45 ^∘ line in the DD-plot. This means that the distributions of the two groups are almost the same.

Fig. 3

DD-plot based on the Mahalanobis depths (+ : failed firms, ∘: non-failed firms)

In summary, the use of DD-plots is likely to separate any firms encountering imminent bankruptcy from financially healthy firms. Accordingly, in the next step, we tried to classify failed firms and non-failed firms via the nonlinear SVM, based on the values of DD on the DD-plot. Because similar trends were observed when we employed different depth measures (i.e., simplicial and Tukey depths), we illustrate the classification results of nonlinear SVM based mainly on the Mahalanobis depth.

4.3 Classifying via the nonlinear SVM

In the context of the corporate bankruptcy classification problem, we introduced nonlinear SVM to a two-dimensional data-set on the DD-plot (called “DD-SVM”). This study employed three kernel functions for nonlinear SVM, including the RBF, polynomial, and MLP kernel functions. The effectiveness of SVM techniques depends on the proper selection of a kernel function, the parameters of the selected kernel function, and the soft margin parameter C.

For example, there are two parameters associated with the RBF kernels, namely, a tuning parameter σ and C. In general, it is not known beforehand which values of C and σ are the best for any given problem. Consequently, some kinds of model selection approaches must be employed. After conducting a grid-search for the training set according to an approach by Erdogan [9], the best combination of σ and C values was selected by the search algorithm with exponentially growing sequences using five-fold cross-validation such as σ ∈{2⁻¹⁵,2⁻¹³,…,2¹,2³} and C ∈{2⁻⁵,2⁻³,…,2¹³,2¹⁵}. For the first year prior to bankruptcy (or the first year of survival), Table 3 shows that the optimal values of the parameters (σ,C) in the RBF kernel (even if they are not unique) report a prediction accuracy of 96.55% for the validation set. In the polynomial kernel function, the best prediction performance for the first year prior to bankruptcy (or survival) was obtained at values of d = 3, γ = 2⁻³, and C = 2⁵, yielding a prediction accuracy of 96.55%. For the MLP kernel function, the best prediction results for the first year prior to bankruptcy (or survival) were obtained at parameter (γ ₁,γ ₂,C) values of (2⁻⁵,1,2⁵) and (2⁻⁷,1,2⁷), yielding a prediction accuracy of 91.38%. With the results of our analysis, we confirmed that the prediction performance of nonlinear SVM is sensitive to the kernel parameters and the upper bound C.

Table 3 Grid-search results using the RBF kernel function for the first year prior to bankruptcy or survival

Table 4 summarizes the results of the prediction of corporate bankruptcy in Korea over 10 years. We compared the prediction powers of the DD-SVM with the prediction powers of nonlinear SVM based on 53 financial ratios. Overall, the DD-SVM outperforms nonlinear SVM in terms of prediction accuracy for corporate bankruptcy. As shown in Table 4, it was noted that the prediction accuracy resulting from DD-SVMs drastically decreases after seven years prior to bankruptcy. The RBF kernel of the DD-SVM produces the best prediction performance for 10 years. Prediction powers from the polynomial kernel are comparable with those from the RBF kernel. However, the polynomial kernel requires more hyperparameters (the polynomial degree d) than the RBF kernel, and under-fitting and over-fitting problems can occur when d is poorly selected. Thus, we selected the RBF kernel to compare with other traditional approaches in the prediction of bankruptcy among Korean firms. Figure 4 presents the classification results using the RBF kernel in nonlinear SVM on a DD-plot based on the Mahalanobis depth. Note that all of the SVM analyses were conducted using R kernlab package.

Table 4 Comparisons of prediction accuracy between DD-SVM and SVM

Fig. 4

SVM classification results using the RBF kernel function. Filled circles and triangles represent support vectors ($\vartriangle$: failed firms, ∘: non-failed firms)

We compared the prediction powers of the DD-SVM with other traditional bankruptcy prediction methods. The bankruptcy classifiers considered in the comparison studies are the logistic regression, multiple discriminant analysis (MDA), and artificial neural network (ANN). The logit model in the logistic regression was employed to investigate the relationship between binary response and financial ratios without the assumption of multivariate normality. The regression parameters were estimated using the maximum likelihood (ML) method. The final logit models for 10 years were selected using the stepwise selection method. We introduced MDA to derive a linear combination of 53 financial ratios that best discriminates between failed and non-failed firms. The ANN model in this study employed a three-layer connected back-propagation. Following the approach in Min and Lee [26], after fixing the number of layers to one, we varied the number of hidden nodes (8, 12, 16, 24, 32) at learning epochs of 100, 200, and 300 and then recorded the parameter values that derived the best prediction powers for 10 years. Table 5 reports bankruptcy prediction results for the manufacturing firms for 10 years, along with Type I an Type II errors. The logit and MDA models show poor prediction powers from one to five years prior to bankruptcy. As confirmed in the preliminary analysis, the financial ratios in this study deviate from the multivariate normality assumption, while the MDA performs well under multivariate normality. The DD-SVM outperforms the ANN model, which is a main comparison target from one to six years prior to bankruptcy. Overall, the DD-SVM has the best prediction power among prominent bankruptcy prediction models reviewed in this study. Because of the impact on the domestic economy, Type I error is more important to diagnose in firms at risk for bankruptcy. Remarkably, the DD-SVM shows almost zero Type I errors from one to five years prior to bankruptcy.

Table 5 Comparative analysis of prediction accuracy for corporate bankruptcy

5 Discussion

Because corporate bankruptcy causes heavy damage to the economy, it is essential to diagnose firms at risk for bankruptcy in order to hedge against the financial harm the at-risk firms stand to inflict. We presented a novel hybrid method that combines DD and nonlinear SVM for the prediction of corporate bankruptcy. This study pioneers the application of DD to financial distress prediction. The empirical results demonstrate that the proposed method offers the highest level of accuracy in bankruptcy prediction. By condensing various financial data from the firms through DD metrics, the DD-SVM can be utilized as part of an early warning system for corporate failure. Unless pertinent information about an underlying distribution is available, we strongly recommend the use of our procedures (detailed in Section 4) to establish a prediction model for corporate bankruptcy. The proposed method is expected to help provide a guidance on corporate investment for investors or other potentially interested parties.

Our comparison is based on a data-set with an equal proportion of failed and non-failed firms. In real cases, however, the number of bankrupt firms constitutes only a small portion of the whole population. To avoid modeling bias, which may be caused by the one-to-one matching process, the entire population of both failed and non-failed firms can be used to construct a corporate bankruptcy prediction model. Financial distress is not the only area wherein DD based SVM shows potential for beneficial use. We believe that the DD-SVM can be extended to other managerial applications, particularly classification problems in various areas.